What you want is an ORDERED set. (Or maybe a partially ordered set). More properly, a totally ordered set.

A totally ordered set can surely have a greatest or least element. Take the ORDERED set [0,1].

Some don't. And they don't have to be all that "big" in the intuitive sense. The ordered set (0, 1) with endpoints "removed" gives you an infinite set where, no matter where you point, you can find a bigger number. (If you point at x, then x + (1-x)/2 is bigger).

No problem. You would still need to clarify what you meant by bounded and unbounded if you think of a set that is closed on one end and approaches infinity or negative infinity on the other. Like the domain or range of the square root function. If the set is completely unbounded, I'm not sure what the answer to your original question is...

Try the Wikipedia article on ordinals. Limit ordinals have no greatest element, successor ordinals do. You are confusing the notion of order type with cardinality. The ordered set {1,2,3,4,...,w} where "w" is some element defined to be larger than all the naturals is an infinite set with a last element, because "last" just means "no element of the set is greater than that one". It is, however, obviously the same cardinality as {1,2,3,4,...}.

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